#8096901

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,6,5)(3,8,4,7)(9,10,15,19)(11,13,14,16)(12,18,23,17)(20,24,21,22)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,3,6,4)(2,7,5,8)(9,11,15,14)(10,16,19,13)(12,21,23,20)(17,22,18,24)(28,36)(32,37)(34,40)(38,39);(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,24)(18,19)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39);(1,3,6,4)(2,7,5,8)(9,13,15,16)(10,11,19,14)(12,22,23,24)(17,20,18,21)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,6,3)(2,7,5,8)(9,12)(10,18)(11,21)(13,22)(14,20)(15,23)(16,24)(17,19)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38);(1,5,6,2)(3,8,4,7)(9,14,15,11)(10,16,19,13)(12,21,23,20)(17,24,18,22)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004097);